Domanda del libro di testo
Evaluate each limit.
lim θ→0 (1/(2+sinθ)-1/2)/sin θ
388
views
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 58
Evaluate each limit.
lim x→0+ 1−cos^2x / sin x
Guida verificata passo dopo passo1
Step 1: Recognize that the expression \(1 - \cos^2 x\) can be rewritten using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Therefore, \(1 - \cos^2 x = \sin^2 x\).
Step 2: Substitute \(\sin^2 x\) for \(1 - \cos^2 x\) in the limit expression. The limit now becomes \(\lim_{{x \to 0^+}} \frac{\sin^2 x}{\sin x}\).
Step 3: Simplify the expression \(\frac{\sin^2 x}{\sin x}\) by canceling one \(\sin x\) from the numerator and the denominator. This simplifies to \(\sin x\).
Step 4: Evaluate the limit \(\lim_{{x \to 0^+}} \sin x\). Since \(\sin x\) is continuous at \(x = 0\), you can directly substitute \(x = 0\) into \(\sin x\).
Step 5: Conclude the evaluation by noting that the limit of \(\sin x\) as \(x\) approaches 0 from the positive side is simply \(\sin(0)\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the expression as x approaches 0 from the positive side (0+). Understanding limits is crucial for evaluating functions that may not be directly computable at specific points.
Video consigliato:
05:50
One-Sided Limits
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the given limit, we encounter cos^2(x) and sin(x), which require knowledge of their properties and behaviors, especially near critical points like x = 0, where they can exhibit specific limits and values.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in simplifying complex limit problems involving trigonometric functions.
Video consigliato:
5:50Power Rules
Pratica correlata
Domanda del libro di testo
Determine the interval(s) on which the following functions are continuous; then analyze the given limits.
f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)
435
views
Domanda del libro di testo
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 3(x − 4)√x + 5 / 3 − √x + 5
281
views
Domanda del libro di testo
Evaluate each limit.
lim x→0 e^4x−1 / e^x−1
411
views
Domanda del libro di testo
Evaluate each limit.
lim x→e^2 ln^2x−5 ln x+6 lnx−2
396
views
Domanda del libro di testo
A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
lim x→−2^− f(x)
321
views